AD-A128 933 CALCULAT ION OF WAVE St4OALINO W IH 0 SSIPAT ON OVER 1/NEARSHORE SANDSMU COASTAL ENG INEER INO RESEARCH CENTERFORT RELVOIR VA R J HALLERMEIER MAR 83 CERC-CETA-83-

UNCLASSIFIED 0G8/3 NI

EEE~h~hhThE

1 1 1 6A1

I25 =6

MICROCOPY RESOLUTION TEST CHARTNAMlCNAL BUREAU OF STANDARDS- 1963-A

CETA 83-1

Calculation of Wave Shoaling WithDissipation Over Nearshore Sands

byRobert J. Hollermeier

COASTAL ENGINEERING TECHNICAL AID NO. 83-1

MARCH 1983

Approved for public release; LKI distribution unlimited. j

U.S. ARMY, CORPS OF ENGINEERS £ -

COASTAL ENGINEERINGW RESARCHCENTER

Xingman BuildingFort Belvoir, Va. 22060

813 (14 08l 003

Reprint or republicationi of any of this materialshall give appropriate credit to the U.S. Army CoastalEngineering Research Center.

Limited free distribution withln the United Statesof single copies of this publication has been made bythis Center. Additional copies are available from:

National Technical Information ServiceATTN: Operations Dieiion5285 Port Royal Road

Springfield, Vi.ginia 22161

Contents of this report are not to be used foradvertising, publication, or promotional purposes.Citation of trade names does not constitute an officialendorsement or approval of the use of such commercial

4. products.

The findings in this report are not to be construedas an official Department of the Army position unlessso designated by other authorized documents.

* I

+ ll+,I i iii' i iiii

UNCLASSIFIEDSECURITY CLASUPICAYIok oP TIs PAGE tvww bt. xwene

4L TIT~E) dASo . CyOREPTOR T UERICOER

Robert J. Hallerueier

0. PERPORINS OftSANIZATION NAME ANID ADDRESS 10. PEMEN?09 PROCT. TAMK

Department of the Army ~2L 0 ~wnuuECoastal Engineering Research Center (CERRE-CP) C15

a Kingman Building, Fort Belvoir, VA 22060It. CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE

Department of the Army March 1983Coastal Engineering Research Center is NGeR OF PAGES

Kin n Building. Fort Belvoir. VA 22060 2'.M NIRtO AENYAM ADORESWj' aNemt GmCrnuebII *Mfe*) ISL SECURITY CLASS. (of ai M tj

UNCLASSIFIED

-14, OSSTRIbUTION TIDR (f*oepi

'~1 Approved for public release; distribution unlimited.

l. SUPPLEMENTARY NOTES

It. KEY WORDS (Calbe. -=#ove" ea& 0080..mp %d lmf reelm

*Agitated sand bed Nonlinear wave propagationBottom friction Rough turbulent flowEnergy dissipation Wave heightNearshore waves Wave shoaling

I* Assraicr elini m ass w omm =9 r Siaml awom*---- This report provides a simplified calculation procedure for nearahore

wave height changes considering the energy dissipated by rough turbulent flowover a strongly agitated bed of quartz sand. All elsmentary wave relation-ships are from linear monochromatic wave theory, but one offet of Includingdissipation is that calculated height changes depend on the absolute aveheight. The general effect of appreciable aiargy lose Is to make field Waveheight relatively constant outside the breaker zone. Rzmple caqwtatossand a calculator program are provided.

OD w am eW Imw oft so a DOME"3ISeaftm" cLFw A1RS I SM71

PREFACE

This report provides a calculation procedure for nearshore shoaliag ofenergetic waves outside the breaker some, Including the appreciable effects ofenergy dissipation over a strongly agitated sand bed. The work reported usaconducted under the U.S. Army Coastal Engineering Research Center's (CUC)Uwurical Modeling of Shoreline Response to Coastal Structures work unit, shoreProtection and Restoration Research Program, Coastal Engineering Area of CivilWorks Research and Developmient.

The present treatment replaces guidance previously provided in CRC FieldGuidance Letter No. 79-04 end CIRC Technical Paper No. 80-8 (Groaskopf, 1960).Those publications Incorrectly recomend the use of friction coefficients forInert boe in computing nearshore wvew shoaling.

The report uas written by Dr. Robert J. N alisrusier, Oceanographer, underthe general supervision of Mr. R.P. Savage, Chief, Research Division, CIRC.

Technical Director of CERC was Dr. Robert W. WhelmA, P.R.

Coents on this publication are Invited.

42

J ~. Approved or publication in accordance with Public Law 166, 79th Congress,approved 31 July 1945, as supplemented by Public Law 172, 88th Congres4 approved 7 November 1963.

Colonel, Corps of EngineersCoinnder and Director

ACCOS~f S

Diet WOSIAL

-,

IU ICONTENTS

PageCONVERSION FACTORS, U.S. CUSTOWAIT TO METRIC (SI). . . . . . . . 5

SiMOLS AND DUZIymIOnS .................... 6

7. IrE ''ON ... . . . . . . . . . . . 7

rI CALXCUATION Pl.OCEDURE . .. . . . . . . . . . . . . ..... * 7

IV APPLCTIa* - 15

LITRATURE CITSD o 6 o v . o o . . . . .. . . 16

APPEDIX CALCULATOR PROGRAM FOR WAVE SHOALING WITH DISSIPATION. .. , 17

I,

-14

4

CORVISIOU FACTOS, U.S. CoSTMWUh TO ITMIC (8I) oIT oF

U.S. customary units of measurement used in thfi report ca be converted tometric (SI) mits as follows:

DMtiply by TO obtaininches 25.4 millimeters- 2.54 centimeterssquare inches 6.452 square centimeterscubic inches 16.39 cubic centimeters

Sfeet 30.48 centimeters

o0.•3048 motors

square feet 0.0929 square moterscubic feet 0.0283 cubic meters

yards 0.9144 asterssquare yards 0.836 square mterscubic yards 0.7646 cubic meters

miles 1.6093 kilometerssquare miles 259.0 hectares

knots 1.852 kilometers per hour

acres 0.4047 hectares

f oot-pounds 1.3558 newton asters

illibars 1.0197 x 10- 3 kilograme per square centimeter

ounces 28.35 grass

pounds 453.6 grams0.4536 kilogram

ton, long 1.0160 metric tons

ton, short 0.9072 metric tons

degrees (angle) 0.01745 radians

Fahrenheit 4agrees 5/9 Celsius degrees or KllvinS1

I1b obtas Clslus (C) temperature readings from Fahrenheit (1) readings,use formula: C - (5/9) (1 -32).

To obtain Kelvin (K) readings, use formula: K - (5/9) (F -32) + 273.15.

I-5

t t I III I ..

m8simau *m UaWZUZTIO

c - LIT wave celerity

D median send grain dimester

d moan water depth

d maximum water depth for sand bed agitation by waves

I average energy dissipation rate

lfo energy dissipation coefficient for rough turbulent flow overstrongly agitated sand bed

* acceleration due to gravity

II wave height

equivalent wave height In deep water Ignoring refraction

X8 shoaling coefficient in linear wave theory

L wavelength

Lo - T2/2w wavelength in deep water

a ratio of group velocity to wave celerity

P average wave energy flux

T wave period

X wave propagation distance

horizontal amplitude of near-bed fluid orbit

P fluid density

Additional subscripts

j value at location where wave condition is to be predicted

a value at geoetric mean water depth for region of interest

1 value at location where wave condition is defined Initially

1- case where dj is greater or les than d1

6

CALCULATION OF WAVE SHOALING W17 DISSIPATION

Robrt J. 14teme

i. INTRODUCTION

* ;As waves propagate toward breaking in shallow water, their attributes aretransformed by effects of water depth and bottom features. These nearshoretransformations are crucial in the interpretation of wave measuresents and inthe ptediction of sediment transport. Prior to wave breaking, appreciable waveenergy can be dissipated by friction between the oscillatory water uotion andthe nearshore bottom, especially where waves cause strong agitation of bottomsediments. This report considers such frictional dissipation.

Ocean waves can be represented most adequately as distributions of propa-gating energy with respect to frequency and direction. Near the shore,extremely energetic waves are observed to constitute a somewhat regular wavetrain approaching along a shore-normal line, with a rather well-defined waveheight and period. According to the Shore Protection Manual (SPM) (U.S. Army,Corps of Engineers, Coastal Engineering Research Center, 1977), an idealizationof demonstrated value in coastal engineering is to represent real waves by acharacteristic height and period, the significant wave condition; this waverepresentation is utilized here.

This report provides a simple calculation procedure defining changes insignificant wave height due to water depth differences combined with bottomfriction at agitated sand beds. The procedure uses linear wave theory betweenthe two depths of interest and a computation of energy dissipation rate at anintermediate water depth. Factors in wave transformation ignored here include:wave direction and complex spectra, currents, surface wave breaking, winds,water viscosity, and bottom percolation and elasticity. The technique is meantfor application only to energetic field wave conditions in fairly shallow waterwith straight, parallel depth contours and relatively fine quartz bottom sands.

Section II presents the calculation procedure for wave height changesconsidering energy dissipation. Section III addresses the application of thisprocedure and includes two example problems: converting nearshore wave measure-ments into equivalent wave heights in shallower and in deeper water. Thereader Is referred to hallermeler (in preparation, 1983) for a detailed sub-stantiation of elements incorporated In the calculation procedure; that refer-ence reviews the empirical basis for the expression giving energy dissipationcoefficient and reports extensive calculated results in clear agreement withmultiple wave measurements at the Coastal Ingineering Research Center's (CMIC)Field Research Facility, Duck, Nor% Carolina, and at other sites.

II. CALCULATION PVOCEDUR

Waves are presumed to travel perpendicular to depth contours with propaga-tion described by small-amplitude (linear) wave theory. Required theoreticalrelations are provided in Section 2.23 of the SPM. Wave and fluid character-stics arising In the wave description are

7

'i.

I I= Iri! .","r ' I

d - local Sean water depth

g - acceleration due to gravity

R - local wave height

L - local wavelength

- " T - wave period

p - fluid density

Wavelength in deep water is Lo - (ST2/2w) and the dimensionless local wave-, length, d/L, is the solution of

"4 d tah2 d d

which is presented in Table C-1 of the SPN. The average wave energy flux perunit crest width is

pg R C n (2)*1 where c - (L/T) is wave celerity and u the ratio of group velocity to wave

n -h4wd/L (32sinh O4W/L)

With no wave refraction, and provided no energy has been added to or removed-from the wave train, it is convenient to express wave height changes by thefactor

H 1 -Ks (4)

no /2n tanh (2wd/L)

where H is equivalent wave height in deep water (ignoring refraction), andKs the shoaling coefficient. The dimensionless quantities n and K areprovided for specific values of (d/Lo) in Table C-1 of the SPK. In situationsconsidered here, a nearshore wave measurement at water depth dl is to beconverted into the corresponding wave condition at another relatively shallowdepth, djV The wave period is presumed constant during propagation so that(di/L o ) and (dj/L o) are known, and equation (4) would give the ratio of waveheights as

- Kg1 (5)

if energy dissipation were to be ignored.

8

Ihergy lost from the wave train due to bottom friction Is treated by meansof a singa dissipation calculation at the geometric mean depth for the regisonof interest

d - V I(6)

Average energy dissipation rate at din, per unit crest width and per unit

length in the propagation direction, 1s lives by

U. - 0.235 p f n (2v &a/T)3 (7)

For rough turbulent flow over a strongly agitated bed of quartz sand, the energydissipation coefficient Introduced in equation (7) is

fee - exp [-5.882 + 14.57 (D/C.)01J ]

Here D is median sand grain diameter at du and u is horizontal ampli-tude of the near-bed fluid excursion arising at du without energy dissipation,so that (2w&U/T) in equation (7) is peak near-bed fluid velocity. According

to linear wave theory,

I i Hu(9)1 n 2 .*nh (2wdm/Lm)

where Um= (RI Km/Ksl) from equation (5).

Energy-conserving linear wave shoaling is combined with computed energydissipation rate into an expression giving a wave height at dj equivalent tomeasured wave height at d1. This expression is a revised form of equation(2):

HR2 80F1 ± i~~x) (10)"]"P a ej n lO

where X is the wave propagation distance between the two water depths (d1and dj), and the upper [lower) sign is used when d1 is greater [les] thand 1 . The conversion given in equation (10) presumes that computed dissipationrate at ds can be considered representative of the entire propagation path.

II11. APPLICATIONS

Besides the explicitly ignored factors affecting nearshore wave transfor-ations, it Is important in applications to consider the requirmeute stated

above on the use of equation (8) for energy dissipation coefficient. Thequartz sand bed mst be strongly agitated by wave action and near-bed flowmst be rough turbulent. Appropriate situations correspond to nearshore fieldwaves with relatively large height and period.

The strength of bed agitation say be Judged usig en ap imte aMpreseM(Ullermeer, 1961, eq. 10) gibing umMm water depth, da, ftr Wve "Ittato

-bow

* j,, * I ;I!

of a quarts sand bed wheu viscous effects are negligible:

da - HT (g/5000 D) 0 5 (11)

If da computed from R1, TI, and D. is much larger than the mxJum waterdepth of interest, the requirement for strong bed agitation may be considered"JJ satief Led.

To asess whether flow is likely to be rough turbulent, another simplecomputation can be performed. incorporating the same approximation for C In

Intermediate water depth (2wd/L near unity) as is utilized in equation (11),fundamental results reviewed in Hallermeier (in preparation, 1983) support

HT > d (metric units) (12)

as an approximate criterion for rough turbulent flow at a strongly agitated bed

of quartz sand. If equation (12) is true according to H1 , TI, and the

mitaxm water depth of Interest, the requirement for rough turbulent near-bed

flow may be considered satisfied.

The following example problems demonstrate the use of the present procedure

In calculating nearshore wave shoaling with energy dissipation due to a stronglyagitated sand Ned.

** * * *********** EXAMPLE PRO*E * *************

GIVEN: At the CERC Field Research Facility, significant wave height exceeding3.5 meters was recorded during three 1981 storms by a Waverider btloy located

in an 18-meter mean water depth. Wave periods associated with these ex-tremely high waves ranged from 9.3 to 14.0 seconds. Nearshore bathymetry atthis site is regularly surveyed to the 9-meter water depth contour, and thewave characteristics at the seaward boundary to the survey region are ofInterest. The shore-normal distance between the water depths is 1800 meters,and the representative sand size for the intervening bottom is D - 0.12millimeter.

FIND: T.ia wave height at di l 9 meters corresponding to H, - 3.5 meters atdl - 18 meters for: (a)T! - 9.3 seconds and (b) T, - 14.0 seconds.

SOLUTION:

(a) For T1 -9.3 seconds and d, - 18 meters

d_ 2wdl. (2w)(18) - 0.1333

1; Tl 9.81 (9.3)2

Table C-1 in the SM gives nj - 0.7570, (aI ) - 0.9160 - Ksi and

dl/L 1 - 0.1694, so that L1 - 106.3 meters. With 81 - 3.5 meters andp - 1026 kilogrm per cubic meter for saltwater, equation (2) becomes

tO

-~ ~ c, n1 j (1026) (9.81L)(3.5)2 106.3 (0.757)

1.33 105 kilogram-meter per scond cubed

t1

L -9.81 (9.3)z .92

and TbeC-1 gives (da/La) - 0.1360, sinh (2irm/Lo) -0.9621, (U3/U~)-0.9378 -Km, and rni - 0.8199. Thus, Ha - (H~ KmeIlsi) - (3.5)(0.9378)/(0.9160)- 3.58 meters and according to equation 639),

HM- 3.58 -1.86 meters

I 2wdm\ 2(0.9621)2 sinh~ La

With Da - 0.12 millimeter, equation (8) is

fz- exp [-5.882 + 14.57 (DM/r.0)0.1941

a exp [-5.882 + 14.57 (1.2 . 10-4/1.86)0.1941 0.0262

so that equation (7) becomes

Em -0.235 (0.235)(1026)(0.0262) r_____3

- 12.5 kilograms per second cubed

Some conditions must also be computed at dj 9 meters, where

d(2w) 9~j. - -0.06665

so that Table C-i gives (Hj/H;) - 0.9779 - K, nj -0.8688, and (d ILj)-

I -0.1108 so that Lj - 81.2 meters. Finally, because d1 ' d, the lower ignin equation (10) is appropriate, and X - 1800 meters yields

-8 (PI - 1m) 8[l.33 . 105 - (12.5)(1800)] 15 qaemtr

i ~ C P c i (1026) (9.81) ..Ll.(0.8688)

-i 3.40 meters

From equation (11), maximum water depth for bed agitation is do -R T1(&/5000D)

0.5 a(3.5)(9.3)[9.81/(5 .103)(0.12 .10-3)10.5 -131.6 meters,

much larger than water depths in the region treated, and the numerical value4 in metric units of (H T I) - 32.6, nearly twice the maximum water depth con-

sidered in meters, so that equation (12) indicates rough turbulent flowthroughout the region. The calculation procedure is suitable for these con-ditions, and the effect of bottom friction on wave shoaling is appreciable,I in that linear wave theory without dissipation would predict a nearshore wave

*1 height of (H1 Ksj/Ksl) - [(3.5)(0.9779)/0.91601 - 3.74 meters, using equation(5).

(b) For T, - 14.0 seconds and d, - 18 meters, (dl/Lo) - 0.05882 so thatn i - 0.8833, (Hj/Hi) - 0.9963 - K81 , (d1/Lj) - 0.1031 and L, - 174.6 metersfrom Table C-1. Equation (2) with H, - 3.5 meters gives

- (1026)(9.81)(3.5)2 (174.6) 8833)8(14.0) (.83

= 1.70 - 105 kilogram-meter per second cubed

At (dm/Lo) - 0.04160 so that (dm/L) - 0.08509, sinh (2wd./L) - 0.5605,(Hm/% ) - 1.057 - Km, and n - 0.9161. Thus, H. - (HI Ksm/Ksi) 3.71 meters,equation (9) is

3.71m -- 3.31 meters

2(0.5605)

equation (8) is

fer M exp {-5.882 + 14.57 (1.2 1 10-4/3.31)0 -1941 - 0.0207

and equation (7) is

- [ 2w(3.31) 3'Em- (0.235)(1026)(0.0207) 14.0

- 16.36 kilograms per second cubed

At dj, (dj/Lo ) - 0.02941 so that (Hj/Hgt) - 1.130 - Kj, nj - 0.9400,(dj/Lj) 0.0706 and Lj - 127.5 meters. Again, the lower sign in equation(10) is used to give

H2 - 8[l.70 . 105 - (16.36)(1800)] . 13.05 square metersi .# (127.a)

(1026) (9.81) (12.5) (0.9400)

H - 3.61 meters

Because T is greater here than in part (a), the requirements for roughturbulent flow at a strongly agitated sand bed are clearly satisfied. Althoughthe computation including friction results in Hj slightly larger than H1in this case, energy dissipation again has an appreciable effect since linearwave theory would predict a nearshore wave height of (H1 Ksj/Kgl) - 3.97 meters,

. from equation (5).

12

i7

*************** , • • , • • • EXAMPLE PROLM 2************

GIVEN: A mathematical model is to be used to staulate storm wave effects forwater depths shoreward of 9 meters at Nags Head, North Carolina, with thethreshold for storm waves taken to be the wave height exceeded 10 percent ofthe time. The wave climate at this site has been defined by several rela-tively complete years of data from a pier-mounted gage located in mean waterdepth of 5.2 meters (Thompson, 1977): wave height exceeded in 10 percent ofthese measurements is about 1.7 meters and the typical wave period for thiswave height is about 8.5 seconds. The shore-normal distance between 5.2- and9-meter water depth is 600 meters; representative sand size for the inter-vening bottom is D - 0.20 millimeter.

FIND: The wave height at dj -9 meters corresponding to H, - 1.7 meters andT1 - 8.5 seconds at d1 .5.2 meters.

SOLUTION: For T,= 8.5 seconds and d1 - 5.2 meters,

1d, 2wdl (2w)(5.2)

Lo - - (9.81)(8.5) - .04610

Table C-1 in SPM gives nj - 0.9074, (HI/) - 1.038 - K.1 and d1 /L1 -0.09002, so that Ll - 57.76 meters. With H! - 1.7 meters, equation (2) is

H2 c 1 n j (1026)(9.81)(1.7)2 (57.76) (0.9074)

8 1 - 1 (8.5)

= 2.24 104 kilogram-meter per second cubed

Since da = d A(5.2)(9) - 6.84 meters

jI.dm (2w)(6.84)

- (9.81)(8.5)2 - 0.06064

and Table C-1 gives (dmn/m) - 0.1049, sinh (2wdm/Lu) = 0.7082, (n.In%) =0.9916 - Ks , and n. - 0.8799. Thus, Ha - (HI Kam/Ks) - (.7)(0.9916)/1.038 - 1.64 meters and according to equation (9),

- HE 1.624 1.15 meters

&M -2 a n 2 d 2(0.7082)

With Dm= 0.20 millimeter, equation (8) is

fen exp (-5.882 + 14.57 (DM/m)0 '1 9 ')

e axp (-5.882 + 14.57 [2 • 10-/1.15] 0.194 ) = 0.0422

13

............................................. . ..

so that equation (7) is

, 0.235 p f - (0.235)(1026)(o.0422) [2i (.s1)

= 6.25 kilograms per second cubed

At dj - 9 meters,

-= (2w)(9) - 0.0798Lo (9.81) (8.5) 2

so that Table C-1 gives (R1/'l) - 0.9551 - Ksj, nl - 0.845, and (dj/Lj) -0.1230 so that Lj - 73.2 meters. Finally, because dj > d, the upper

sign in equation (10) is appropriate, and X - 600 meters yields

R2 8 (PI+ EaX) . 8 (2.24 . 10" + (6.25)(600)] - 2.86 square meters

j P g cj nj (1026)(9.81) (73.2) (0.845)

-= 1.69 meters

To show that the calculation procedure is suitable for these conditions,maximum water depth for bed agitation from equation (11) is

10.5r 9.81"d a - B, T, (1.7)(8.5) L(5 •103)(2 . 10"- )

- 45.3 meters

such larger than water depths in the region treated, and the nmerical value in

metric units of (HIT,) - 14.45, greater than the maximm water depth considered

in meters, so that equation (12) indicates rough turbulent flow throughout the

region considered. The effect of bottom friction is still appreciable for

this relatively low-energy case, in that linear wave theory without dissipation

would provide a wave height according to equation (5) of (KI Kaj/K 5 1 ) -

[(1.7)(0.9551)/1.0381 - 1.56 meters at 9-mster water depth.

With linear wave theory, the height change between two water depths depends

on wave period. Although only linear theory wave relationships are incorpora-

ted in the present calculation procedure, energy dissipation depends both on

wave period and wave height (raised to the power of about 2.5). Thus, the

calculated results have a nonlinear dependence on wave height: the computed

height change between two water depths is affected by the actual value of wave

height.

1"4

This nonlinear aspect Implies that thes computations are not exactlyreversible. Projecting a wave condition offshore without dissipation from aMeasurement site to da can result In a markedly different computed dissipa-tion rate there than if the nominally corresponding waves are projectedonshore to dm. However, the calculation procedure tends to cancel Internallythis effect of nonlinear height dependence.

Reversing Example Problem 2, using H1 - 1.69 meters at dl - 9 meters asthe Initial condition, computed conditions at du Include Cm - 1.24 meters,fen = 0.0406 and Em 7.52 kilograms per second cubed, but at dj - 5.2 metersthe calculated wave height ts 1.67 meters, only 1.8 percent less than the near-shore wave height 6f 1.7 meters originally specified. Using computed finalwave heights in Example Problem 1 as Input conditions, the reverse calculationprocedure gives wave heights in each case only .2. percent less than the specifiedheight of 3.5 meters.

Such slightly irreversible results do not sem too significant for potentialapplications. However, the Appendix to this report provides a calculatorprogram quickly executing the present procedure, making it convenient toexamine results of reverse calculations and to determine a wave condition whichappears optimally consistent with that specified.

IV. SIMOARY

The equations and procedures presented here permit calculation of nearshorewave height changes considering the energy dissipated by rough turbulent flowover a strongly agitated bed of quartz sand. All elementary wave relationshipsare from linear (small-amplitude) wave theory, but one effect of Incorporatingdissipation is that calculated height changes depend on the actual wave height.Example calculations demonstrate the conversion of a nearshore wave conditioninto a corresponding wave height in shallower or deeper water; the presentprocedures are suitable only for field waves of relatively large height andperiod in fairly shallow water. The general effect of energy dissipation isthat nearshore wave height remains more nearly constant outside the breakerzone than linear wave theory would predict.

1

I I I I' I r '1 l l r '

LITUATURK CITUD

G4tOSSICW, V.G , "Calculation of WVa Attemuatin Due to Pietim and Uholio :An Evluation," TV W0-8, U.S. Any, Corps of ngineers, Coastal UgtaeleriftResearch Center, Tort selvoir, Vs., Oct. 1980.

NALLUKMIR, 1.J., "Seaward Limit of Signif leant Sand Transport by Waves: AnAnnual Zonation for Seasonal Profile*," CRIA 81-2, U.S. Army, corps ofEngineers, Coastal Engineering Research Center, Fort Belvoix, Va., Jan. 1981.

BALLEUMR , R.J., "Gravity Wave Dissipation Over Agitated Nearabore Sands,"(In preparation as journal article, 1963).

TSo(DPSO, B.F., "kave Chumte at Selected Locations Along U.S. Coasts,"TR 77-1, U.S. Army, Corps of Engineers, Coastal Engineering Research Center,Fort Delvoir, Va., Jan. 1977.

U.S. AMY, CORPS OF XINURS, COASTAL ENGON)RIG RESEARCH CEMTU, ShoreProtcotion Mvmat, 3d ed., Vols. 1, IT, and III, Stock No. 008-0.22-00113-1,U.S. Government Printing Office, Washington, D.C., 1977, 1,262 pp.

I

161

APPUDIX

CALCLATOR PROGRAM FOR WAVI SUOALIU W17R DISSIPATION

The following four pages document a calculator program executing theprocedure presented in Section 11 and exmpliffted In Section III of this report.This program runs in about 120 seconds on a Hewlett-Packard RP-67 ProgrammablePocket Calculator, employing metric units, KIN logIc, three levels of nestedsubroutines, 17 address labels, 26 storage registers, and 223 program steps.The program could be converted for use with other calculators having differentlogic systems but similar features and capacities.

Equations (1) to (10) are included with an' effective root-finding iterationfor wavelength. Values to be specified In metric units for a calculation are:P, 5, H, TI , d 1 , di, D . , and X. The standard value of & is 9.81 meters persecond squared, and the value of p for seawater may be taken as 1026 kilo-grams per cubic meter; for freshwater, p is about 1000 kilosrams per cubicmeter, but common situations might not constitute the requisite rough turbu-lent flow over a strongly agitated bed. Satisfaction of these requirements,related to equations (11) and (12), remains to be considered external to thecalculator program.

'7

-- Il

ft o= If -d * 'Oa g c IV~vjs = 6 tEASm Emm a'~ v s AT T SE

BobI&#" 4LS,&" W*'ftcj opTE LWMAftt~ Ls\y - r TH M momn~ r

mom w-mr, mesm AT '-OAe wA- wx Rm

'NE II'flA %Silf I'iEs-AI'V 3omAItp FOE IVW&F~as1 IC A.VSU Puewo*6& "ilfT~v~~ vbve

T'im ME,A t~iog VFOAI 't@ CAiuIDfamsmyw'T~ ^tnq~y 3*s M SgvmFw.. %A SC69CI -WVC~X

hqomok~~~~ IV -W14l T16PpL a sk-OA O mOIg IMMrL1, S E l". V . ~3Rboija vwC10 Men" US

wti% mL~r-lVI~gPS fovMIOWOM %tnMOoss st me'~f f*takau.lm/pe

th0wa SlO-Al 6AMv. "eANs- is~e m u. ONwu vm WS

03OU0 OOLSO6 TELI 5LA~J TOC VMVO D (104 Pft

@pift3*S ti W oWA F SM WDeS$ IA M " W*P si~ IS VALI h ' QoS

u g)e- fWEEGW dt WSWEwAftE 0 ISPS&Sb coftsTs %fl14 "ANDIMLY

SHSI*- -WfH*ur 09AKIO& ASK W*MASD MS- %&VOCIS& MANV

Um'r Eamarun Sow

~ oWIL1G Wlm*6AU 5WINt GN04V an

eon -~ nv a wvut $ W W m u o 11 :- _ _ _ _ _ _ _ _ _ _own

~~^f go~*L s.DS5~ &. - =J

"Pimo m*A"s - 'w

tog.f vkimwr w .w W]'i JBNA Wkl D W hv%*"W hi

GoMX aAN Manu Wi

mtw-*i m -u

_ _ __ __ _ Qm DMMEKts

* mo __ __ __ _ __ __ _ __ __ _ _____o~ Gf l ]

_ _ __ _ __ _ IM CIw fmoi S d I w oomS vSSftmV ga usco c i

_________________M UDc~c

_________________m an ~ Jc

%A fta~ I1PRP won& OF ma tCD

_oo esm EDm mtmO w"#04.~h~ --a"

&INM b tv~nw. a-&

- ~ ~ r f~in I'u~IIIE

___~vtt - uw mvmv voss

UCMIg" n.b

34J

120

1XL f3 _ _ _ _ _ _ 34

~s +gAIO as) -0

govA~om (2) (43 -

IC- - - - It

it" - - -&&

so. . .4 vSI~. 1600 &1f 64UIL. ~~. IL :-~'U.

0 j b a. so ! ba .

a a 0 "a u

6a 6 " aa .4* 0~

. W a0 .a.9 10 1 V

0421, 1 0 0,

66 .4!

04 6, C4*a .0

.12 V111 3 ZIA4~

so4 Z4~ 0. .

EUC C 6.4

a too0 20 -O

4 6.4.01 w 6.4. 64

96 a .

Iwo .0 -"04..5.a.

v% A4"I . .

0 f" I An

0i Akj8 14.. " I i I U .